1. Techniques for Finding Derivatives
Lesson 4.1

2. Limitations of the Definition
Recall our use of the definition of the derivative
This worked OK for simple functions
Becomes unwieldy for other common functions
Higher degree polynomials
Trig functions

3. Other Ways to Represent The Derivative
Previous chapter
For a function f(x), we used f '(x)
To show derivative taken with respect to a variable
When y is a function of x shows "derivative of y with respect to x"
Other representations

4. Constant Rule Given f(x) = k Then When we evaluate this we get
A constant function
Then
When we evaluate this we get
We conclude when f(x) = k
f '(x) = 0
How does this fit with our understanding that the derivative is the graph of the slope values?

5. Power Rule Consider f(x) = x3
Use the definition to determine the derivative.
Now let h → 0
f(x) = x3
f '(x) = 3x2
What pattern do you see?

6. Power Rule For f(x) = xn Then With any real number n
Decrease the exponent by 1
Multiply the function by the exponent

7. Constant Times A Function
What happens when we have a constant times a function?
Example
The rule is So

8. Sum Or Difference Rule
Consider a function which is the sum of two other functions
Example :
The derivative of f(x) is
The derivative of the sum is the sum of the derivatives

9. Try It Out
Apply all these rules to take the derivatives of the following functions.

10. Marginal Analysis
Economists use the word "marginal" to refer to rates of change.
When we have a function which represents
Cost
Profit
Demand
Then the marginal cost (or profit, or demand) is given by the derivative

11. Marginal Analysis
When the sales of a product is a function of time t = number of years
What is the rate of change or the marginal sales function?
What is the rate of change after 3 years?
After 10 years?

12. Assignment Lesson 4.1A Page 248 Exercises 1 – 45 odd Lesson 4.1B